Arbitrary Constants in Mechanical Problems. 4-95 



Finally, if in the equations 



d a — [b, a] -TV- d t— [c, a] -7— */ £ -f &c. = 0, 



the values of —rj- dt, —, — dt, &c. are substituted from the 

 db dc 



equations which constitute the 1st theorem, viz. 

 — — dt = (a, b) da + {c,b)dc -f (c,b) de +&c. 



— — dt = (a, c) d a + (b, c) d b + (e, c) d e + &c, 



CL C 



the resulting equation must be identically equal to zero; and 

 equating to zero separately the coefficients of da, db, de, &c. 

 in this equation, I obtain the equations 



1 = [b, a~] (a, b) -f [c, a] {a, c) + [a, c] (a, e) + &c. (10.) 



[c, a] (b, c) + [c, a] (b, e) + &c. = (11.) 



[b,a] (c,b) -f [c,a] {c,e) +&c. = (12.) 



These equations are given by M. Cauchy in Liouville's 

 Journal de Mathematiques, 1837, p. 409; but the proof which 

 M. Cauchy has there offered is in my opinion .by no means so 

 .simple as the foregoing. These equations serve to connect 

 the quantities [b, a], \c, a~], &c. with the quantities (b, a), (c, a), 

 &c, so that by elimination the one may be deduced from the 

 other, and they serve to show that these quantities \b,a~], [c,a~\, 

 &c. are also constant, and independent of the time. 



The reasoning which is required to establish the principal 

 theorems I, II and III, is the same in the case of only one vari- 

 able x, and of one differential equation, with two arbitrary con- 

 stants a and b tf x dV dR 



4- -1 = 



df : dx T dx 



as in the more general case of three variables x, y, z, with 

 six arbitrary constants. 



I have written this short paper under the impression that 

 the proof here given of Lagrange's theorem (Theorem I., 

 p. 4-94-,) and the proof given here of equations (10), (11), and 

 (12), have not been presented before; but these methods are 

 extremely simple that it is by no means unlikely that they 

 have occurred to others who may also have sought to bring 

 this important theory within a narrower compass and within 

 the reach of more elementary considerations than those 

 hitherto employed. 



